Mini-Workshop: Mathematics of Solvency

Abstract

The mini-workshop Mathematics of Solvency, organised by Freddy Delbaen (Zurich), Karl-Theodor Eisele (Strasbourg) and Christian Hipp (Karlsruhe) was held during the week February 10th–February 16th, 2008. The 15 participants from academia and practice – most of whom came from supervision authorities – discussed the foundation of mathematical concepts for risk and risk measurement as well as their consequences for various decision or control activities in financial services. Such concepts are of major importance for the definition of a necessary solvency capital – which takes into account diversification effects – for banks, investors, and insurers, from the perspective of supervision, of risk management, of rating, and of capital allocation. They help industry to develop consistent foundations for dealing with insurance and financial risks.

The investigation of risk concepts in the multi period setting leads to new mathematical questions in the theory of stochastic processes. A sensible definition of a risk measure requires the property of super-additivity with respect to different risks as well as a submartingale property with respect to time. Such a definition leads to a theory of non-linear pricing of risks, as well as to a better understanding of the time evolution of risk assessment. These two concepts were not treated in traditional finance.

The consistency of risk measurement over different time intervals leads to monotonicity properties of convex functionals. The representation of these time consistent valuations is related to backward stochastic differential equations and difference equations, respectively. The information available to agents is modeled with filtrations and the difference in information leads to problems known as enlargement of filtrations.

In the talks, methods and concepts from various fields were used: stochastic processes, game theory, functional analysis, and stochastic control. In addition, experience with existing and operating real world risk valuation concepts has been reported and discussed.